Heat equation in vector bundles with time-dependent metric

Journal of the Mathematical Society of Japan

Heat equation in vector bundles with time-dependent metric

Dedicated to the memory of Professor Kiyosi Itˆo on the occasion of the 100th anniversary of his birth

By Robert Philipowski and Anton Thalmaier

(Received xx xx, 2015) (Revised xx xx, 2015)

Abstract. We derive a stochastic representation formula for solutions of heat-type equations on vector bundles with time-dependent Riemannian metric over manifolds whose Riemannian metric is time-dependent as well.

As a corollary we obtain a vanishing theorem for bounded ancient solutions under a curvature condition. Our results apply in particular to the case of differential forms.

1. Introduction

In his talk at the ICM in Stockholm 1962 [12], Professor Kiyosi Itˆo’s showed that the Levi-Civita parallel translation of tensors on a Riemannian manifold makes perfect sense along the trajectories of a Brownian motion. Several mathematicians, among them Paul Malliavin, James Eells and David Elworthy, understood quickly the significance of this construction and took up the new ideas [4, 5, 16, 17]. It turned out to be the starting point of a new mathematical field, stochastic differential geometry, born from a combination of ´Elie Cartan’s method of moving frames and Kiyosi Itˆo’s theory of diffusion processes. The first applications, fully in the tradition of Bochner’s method, aimed at cohomology vanishing theorems under positivity conditions [13, 16, 6, 8, 7]. The idea is to use a stochastic representation of harmonic forms, or more generally of solutions of the heat equation on differential forms, in terms of a certain transport along Brownian motion which can be estimated in terms of curvature. It is the goal of this note to extend these ideas to the setting of Riemannian manifolds evolving under a geometric flow.

LetM be ad-dimensional differentiable manifold equipped with a family (g(τ))τ∈[T₁,T₂]

of Riemannian metrics depending smoothly on τ, and let E be a k-dimensional vector bundle overM, also equipped with a family (g^E(τ))τ∈[T₁,T₂] of Riemannian metrics de- pending smoothly onτ. Let∇(τ) be the Levi-Civita connection ofg(τ), and let ∇^E(τ) be a covariant derivative onE which is compatible with g^E(τ) and which also depends smoothly on τ. TheBochner Laplacian (or connection Laplacian) ∆^τ with respect to

∇^E(τ) andg(τ), acting on smooth sectionsϑofE, is defined as the trace (with respect tog(τ)) of the second covariant derivative (with respect to∇^E(τ) and∇(τ)) ofϑ.

2010 Mathematics Subject Classification. Primary 58J65; Secondary 53C44.

Key Words and Phrases. Martingale, heat equation, vector bundle, geometric evolution, Ricci flow, Bochner Laplacian.

One is often not only interested in the Bochner Laplacian, but more generally in oper- ators allowing a Weitzenb¨ock type decomposition with respect to the Bochner Laplacian of the form

∆^τ− R^τ,

where (R^τ)τ∈[T₁,T₂] is a family of symmetric (with respect to g^E(τ)) endomorphisms ofE, depending smoothly onτ.

The most important example is presumably E = Λ^pM, the bundle of differential p-forms onM, equipped with the family of Weitzenb¨ock operators

R^τ = ∆^τ−^τ,

where^τ denotes the Hodge-de Rham Laplacian on Λ^pM, see e.g. [11, Section 7.1]. In particular, if p = 1, R^τ equals the Ricci tensor with respect to g(τ) considered as an endomorphism of Λ¹M =T^∗M, see e.g. [11, Corollary 7.1.4].

Extending the seminal ideas of Professor Itˆo [12] to the case of time-dependent geom- etry we prove in this paper a stochastic representation formula for solutions of the heat type equation

(1.1) ∂ϑ

∂τ = 1

2(∆^τ− R^τ)ϑ

and apply it to prove a vanishing theorem for bounded ancient solutions under a positivity condition on

R^τ−∂g^E

∂τ .

In the case of 1-forms this condition means that the metric ofM evolves under uniformly strict super Ricci flow.

2. Stochastic representation formula

We denote byR(τ, y) the lowest eigenvalue ofR^τ_y− ^∂g_∂τ^E(τ, y)#g^E(τ)

. Here the super- script^#gE(τ)means that using the metricg^E(τ) we regard^∂g_∂τ^E as an endomorphism ofE.

Remark 2.1. Note that ifE =T^p,qM := (T M)^⊗p⊗(T^∗M)^⊗q is a tensor bundle overM andg^E(τ) the canonical metric induced fromg(τ) on the base manifoldM, we have for allv1, . . . , vp ∈TyM andα1, . . . , αq∈T_y^∗M,

∂g^E

∂τ (τ, y)

^#gE(τ)

(v1⊗. . .⊗vp⊗α1⊗. . .⊗αq)

=

p

X

i=1

v1⊗. . .⊗vi−1⊗ ∂g

∂τ(τ, y) #g(τ)

vi⊗vi+1⊗. . .⊗vp⊗α1⊗. . .⊗αq

−

q

X

j=1

v1⊗. . .⊗vp⊗α1⊗. . .⊗αj−1⊗ ∂g

∂τ(τ, y)

(α^#_j,·)⊗αj+1⊗. . .⊗αq. (2.1)

The reason for the minus sign in formula (2.1) is as follows. Recall that forα, β∈T_y^∗M,

∂g^T∗M(τ, y)

∂τ (α, β) = ∂g(τ, y)(α^#g(τ), β^#g(τ))

∂τ =−

∂g(τ, y)

∂τ

(α^#g(τ), β^#g(τ)) which is a consequence of

0 = ∂

∂τα(·) = ∂

∂τg(τ, y)(α^#g(τ),·) =

∂g(τ, y)

∂τ

(α^#g(τ),·) +g(τ, y)

∂α^#g(τ)

∂τ ,·

.

In the special case of the (backward/forward) Ricci flow ^∂g_∂τ =±Ricg(τ), we thus find ∂g^E

∂τ (τ, y)

^#gE(τ)

(v1⊗. . .⊗vp⊗α1⊗. . .⊗αq)

=±

p

X

i=1

v1⊗. . .⊗vi−1⊗Ric(vi,·)^#⊗vi+1⊗. . .⊗vp⊗α1⊗. . .⊗αq

∓

q

X

j=1

v1⊗. . .⊗vp⊗α1⊗. . .⊗αj−1⊗Ric(α^#_j ,·)⊗αj+1⊗. . .⊗αq. We now fix x∈M and let X = (Xt)0≤t≤T₂−T₁ be a (g(T2−t))0≤t≤T₂−T₁-Brownian motion onM starting atx[1, 3, 9, 10, 14, 15, 18]. Throughout the paper we assume that it cannot explode (see [14, 18] for sufficient criteria). We denote by//^E_0,t:Ex→EXt the parallel transport inE alongX. This is a random isometry which will be defined in the proof of Lemma 5.1.

Theorem 2.2 (Stochastic representation formula). Assume that R is bounded from below, and let Φt:Ex→Ex be the solution to the (random) ODE

(2.2) dΦt

dt =−1

2Φt(//^E_0,t)⁻¹ R^T_X²_t^−t− ∂g^E

∂τ (T2−t, Xt)

^#gE(T−t)! //^E_0,t

with initial valueΦ0= IdEx. (This is a linear ODE, hence its solution cannot explode.) Then any bounded solution ϑ: [T1, T2] → Γ(E) to the heat type equation (1.1) has the stochastic representation

(2.3) ϑ(T2, x) =E

ΦT₂−T₁(//^E_0,T2−T1)⁻¹ϑ(T1, XT₂−T₁) . Moreover, the following estimate holds:

(2.4) |ϑ(T2, x)|g^E(T₂)≤E

"

exp −1 2

Z T₂−T₁

0

R(T2−s, Xs)ds

!

|ϑ(T1, XT₂−T₁)|g^E(T₁)

# . The key ingredient to the proof of Theorem 2.2 is the following proposition whose proof is given in Section 5.

Proposition 2.3. Let ϑ: [T1, T2] → Γ(E) be any smooth time-dependent section of E (not necessarily bounded, and not necessarily a solution of Eq. (1.1)). Then the

Ex-valued stochastic process

(2.5) Nt:= Φt(//^E_0,t)⁻¹ϑ(T2−t, Xt) satisfies

dNt=−Φt(//^E_0,t)⁻¹ ∂ϑ

∂τ −1 2

∆^T2−t− R^T_X²_t^−t ϑ

(T2−t, Xt)dt + Φt

d

X

i=1

(//_0,t^E )⁻¹∇_U^E,T_tei^2−tϑ(T2−t, Xt)dBⁱ_t,

where(Ut)0≤t≤T₂−T₁ is the(g(T2−t))t≥0-horizontal lift ofX with respect to an arbitrary initial frameU0∈O_x^g(T2)(M)(see Eq.(5.2)below) and(Bt)0≤t≤T₂−T₁ is the correspond- ing anti-development (which is a standardR^d-valued Brownian motion).

Remark 2.4. In the special case where E is a tensor bundle over M and g^E(τ) is the usual extension of g(τ), Proposition 2.3 is due to Chen et al. [2, Eq. (3.7)]. Our Proposition 2.3 is considerably more general in the following two respects:

– It is not restricted to tensor bundles, but holds on arbitrary vector bundlesEoverM. – The metrics onE need not be related in any way to the metrics onM.

For the proof of Theorem 2.2 we also need the following lemma:

Lemma 2.5. For allt∈[0, T2−T1] we have (2.6) |Φt|_g^E_(T2),g^E_(T2)≤exp

−1 2

Z t

0

R(T2−s, Xs)ds

.

Proof. Fixv∈Ex and letf(t) :=|Φ^∗_tv|²_gE(T₂). Then Eq. (2.2) and the definition ofR imply thatf^′(t)≤ −R(T2−t, Xt)f(t) and hence

f(t)≤exp

− Z t

0

R(T2−s, Xs)ds

f(0), i.e.

|Φ^∗_tv|_g^E(T₂)≤exp

−1 2

Z t

0

R(T2−s, Xs)ds

|v|_g^E(T₂).

Since|Φ^∗_t|_g^E(T₂),g^E(T₂)=|Φt|_g^E(T₂),g^E(T₂)this gives the claim.

Proof of Theorem 2.2. Sinceϑsatisfies Eq. (1.1), Proposition 2.3 implies that the process N defined in (2.5) is a local martingale. Since moreoverϑand Φ are bounded (the latter by Lemma 2.5 and the assumption that Ris bounded from below), N is seen to be a true martingale. Taking expectations ofN att= 0 andt=T2−T1yields (2.3).

The estimate (2.4) follows from (2.3), (2.6) along with the fact that //^E_0,T2−T1 is an isometry from (Ex, g(T2)) to (EXT2−T1, g(T1)).

3. Solution flow domination

Letf be a bounded solution to the scalar heat-type equation

(3.1) ∂f

∂τ = 1 2

∆^g(τ)− R(τ,·) f

withf(T1,·) =|ϑ(T1,·)|_g^E(T₁). Then Eq. (2.2) reduces to the scalar ODE dΦt

dt =−1

2ΦtR(T2−t, Xt) with initial value Φ0= 1. It follows that

Φt= exp

−1 2

Z t

0

R(T2−s, Xs)ds

,

so that Theorem 2.2 implies (3.2) f(T2, x) =E

"

exp −1 2

Z T₂−T₁

0

R(T2−s, Xs)ds

!

|ϑ(T1, XT₂−T₁)|_gE(T₁)

# .

As a consequence we can reformulate estimate (2.4) as follows:

Theorem 3.1 (Solution flow domination). We have

|ϑ(T2, x)|_g^E_(T2)≤f(T2, x).

4. Vanishing theorem for bounded ancient solutions

Theorem 4.1(Vanishing theorem). Let(g(τ))−∞<τ≤T₂ be an ancient family of Riemannian metrics such thatRis bounded from below on (−∞, T2]×M and

(4.1) lim inf

T₁→−∞E

"

exp −1 2

Z T₂−T₁

0

R(T2−s, Xs)ds

!#

= 0

for allx∈M (xenters condition (4.1)as the starting point of the Brownian motionX).

Then every bounded ancient solutionϑ: (−∞, T2]→Γ(E)to the heat type equation (1.1) vanishes.

Proof. It clearly suffices to prove that ϑ(T2, x) = 0 for all x∈ M. By (2.4), for all T1∈(−∞, T2],

|ϑ(T2, x)|g^E(T₂)≤E

"

exp −1 2

Z T₂−T₁

0

R(T2−s, Xs)ds

!#

sup

y∈M

|ϑ(T1, y)|g^E(T₁).

LettingT1→ −∞, the claim follows from (4.1) and the boundedness ofϑ.

Remark 4.2. Trivially condition (4.1) holds if R(τ, y) ≥ C > 0 for all (τ, y) ∈ (−∞, T2]×M. In the special case of 1-forms, this means that the metric ofM evolves

under uniformly strict super Ricci flow, i.e.

∂g

∂τ + Ricg(τ)≥C

for someC >0. Here we used that in this case (see Remark 2.1 above) forα∈T^∗M, R^τ(α) = Ricg(τ)(α^#,·) and ∂g^E

∂τ (α) =− ∂g

∂τ

(α^#,·).

Remark 4.3. Since the endomorphisms R^τ may depend on ϑ, our results can also be applied to nonlinear equations. As observed by Chen et al. [2] such nonlinear equations arise naturally in the context of geometric flows such as the Ricci flow or the mean curvature flow.

5. Proof of Proposition 2.3

To keep notation simple, we assume in this section without loss of generality that T1= 0. Moreover we writeT instead ofT2. We need the following lemma.

Lemma 5.1 (cf. [2, Proposition 2.1] for the case that E is a tensor bundle and ϑ independent of time). The Ex-valued process

N˜t:= (//^E_0,t)⁻¹ϑ(T−t, Xt) (where//^E_0,t:Ex→EXt is defined in the proof ) satisfies

dN˜t=−(//^E_0,t)⁻¹ ∂ϑ

∂τ −1

2∆^T−tϑ+1 2

∂g^E

∂τ

^#gE(T−t) ϑ

!

(T−t, Xt)dt

+

d

X

i=1

(//^E_0,t)⁻¹∇^E,T−t_Utei ϑ(T −t, Xt)dBⁱ_t.

Proof. Letπ:F(E)→M the frame bundle ofE (so that forx∈M the fiberF(E)x

is the set of linear isomorphisms fromR^k toEx). Moreover letF(M)×MF(E) be the product of the fiber bundlesF(M) andF(E), i.e.,

F(M)×M F(E) =

(u, ψ)∈F(M)×F(E)

πu=πψ .

Note that for all (u, ψ)∈F(M)×MF(E) we have the following canonical identification:

(5.1) T(u,ψ)(F(M)×M F(E))≃

(X1, X2)∈TuF(M)×TψF(E)

π∗X1=π∗X2 . Forτ∈[T1, T2] andi∈ {1, . . . , d} we define thei-th standard horizontal vector field

H^τ_i =H^∇(τ),∇_i ^E(τ)

on F(M)×M F(E) with respect to ∇(τ) and ∇^E(τ) as follows: In the sense of the identification (5.1), for (u, ψ) ∈ F(M)×M F(E) the first component of H_i^τ(u, ψ) is the ∇(τ)-horizontal lift of uei to TuF(M), and the second component is the ∇^E(τ)-

horizontal lift ofuei to TψF(E).

Let now Ψ0 be an arbitrary element of O_x^gE(T)(E), and let (Ut,Ψt)0≤t≤T be the solution to the Stratonovich SDE

d(Ut,Ψt) =

d

X

i=1

H^T−t_i (Ut,Ψt)∗dB_tⁱ

+1 2

∂g

∂τ(T−t)

#g(T−t)

◦Utdt +1

2 ∂g^E

∂τ (T−t)

^#g(T−t)

◦Ψtdt (5.2)

with initial value (U0,Ψ0). Note that by constructionπUt=πΨt=Xtfor allt∈[0, T].

In addition, we haveUt∈O_x^g(T−t)(M) and Ψt∈O_x^gE(T−t)(E) for all t∈[0, T]. We then define

//^E_0,t:= Ψt◦Ψ⁻¹₀ .

Similarly to [11, Section 2.2] we define the scalarization ofϑas the map ϑ˜: [0, T]×F(E)→R^k

given by

ϑ(τ, ψ) :=˜ ψ⁻¹ϑ(τ, πψ).

Clearly, ˜Nt= Ψ0ϑ(T˜ −t,Ψt) so that dN˜t= Ψ0dϑ(T˜ −t,Ψt)

=−Ψ0∂ϑ˜

∂τ(T−t,Ψt)dt+1 2Ψ0

d

X

i=1

(H^T−t_i )²ϑ(T˜ −t, Ut,Ψt)dt

+1 2Ψ0

∂g^E

∂τ

^#gE(T−t)

◦Ψt

!

ϑ(T˜ −t,Ψt)dt

+ Ψ0 d

X

i=1

H^T_i^−tϑ(T˜ −t, Ut,Ψt)dB_tⁱ.

The claim now follows from Lemma 5.2, Lemma 6.1 and Corollary 6.3 below.

Lemma 5.2. For allτ ∈[0, t] and allψ∈F(E), ∂g^E

∂τ

^#gE(τ)

◦ψ

!

ϑ(τ, ψ) =˜ −ψ⁻¹ ∂g^E

∂τ

^#gE(τ)

ϑ(τ, πψ).

Proof. We have ∂g^E

∂τ

^#gE(τ)

◦ψ

!

ϑ(τ, ψ) =˜ d ds

s=0

ϑ˜ τ, ψ+s ∂g^E

∂τ

^#gE(τ)

◦ψ

!

= d ds

s=0

ψ+s ∂g^E

∂τ

^#gE(τ)

◦ψ

!⁻¹

ϑ(τ, πψ)

=ψ⁻¹ d ds

_s=0

IdEπψ+s ∂g^E

∂τ

^#gE(τ)!⁻¹

ϑ(τ, πψ)

=−ψ⁻¹ ∂g^E

∂τ

^#gE(τ)

ϑ(τ, πψ),

as claimed.

Proof of Proposition 2.3. We have Nt= ΦtN˜t. Using Lemma 5.1 this implies dNt= (dΦt)(//^E_0,t)⁻¹ϑ(T−t, Xt) + ΦtdN˜t

=−1

2Φt(//^E_0,t)⁻¹ R^T_X^−t_t − ∂g^E

∂τ

^#gE(T−t)!

ϑ(T−t, Xt)dt

−Φt(//^E_0,t)⁻¹ ∂ϑ

∂τ −1

2∆^T−tϑ+1 2

∂g^E

∂τ

^#gE(T−t) ϑ

!

(T −t, Xt)dt

+ Φt d

X

i=1

(//^E_0,t)⁻¹∇^E,T−t_Utei ϑ(T−t, Xt)dB_tⁱ

=−Φt(//^E_0,t)⁻¹ ∂ϑ

∂τ −1

2 ∆^T−t− R^T−t_Xt ϑ

(T −t, Xt)dt + Φt

d

X

i=1

(//_0,t^E )⁻¹∇^E,T_Utei^−tϑ(T−t, Xt)dB_tⁱ,

as claimed.

6. Appendix

In this appendix we fixτ∈[0, T] and, to simplify notation, suppress it in the sequel.

Lemma6.1(cf. [11, Proposition 2.2.1]). Letϑbe a section ofE,x∈M,ψ∈F(E)x, ξ∈TxM andξ¯the ∇^E-horizontal lift ofξ toTψF(E). Then

ξ¯ϑ˜=∇]^E_ξϑ(ψ).

Proof. Let (ψt)t∈R be a horizontal curve (with respect to ∇^E) in F(E) with ψ0 =ψ and ˙ψ0= ¯ξ, and letxt:=πψt(so thatx0=xand ˙x0=ξ). Then

//0,t:=ψtψ⁻¹₀ :Ex→Ext

is the parallel transport (with respect to∇^E) along the curve (xt)t∈R. Consequently, ξ¯ϑ˜= d

dt _t=0

ϑ(ψ˜ t)

= d dt

t=0

ψ_t⁻¹ϑ(πψt)

=ψ⁻¹₀ d

dt t=0

//⁻¹_0,tϑ(xt)

=ψ⁻¹∇^E_ξϑ

=∇]^E_ξ ϑ(ψ),

as claimed.

Lemma6.2(cf. [11, Eq. (2.2.3)]). For(u, ψ)∈F(M)×MF(E)andi, j∈ {1, . . . , n}

we have

(HiHjϑ)(u, ψ) =˜ ψ⁻¹Hessϑ(uei, uej).

Proof. As in the proof of Lemma 6.1 we obtain (Hjϑ)(u, ψ) =˜ ψ⁻¹∇^E_uejϑ

for all (u, ψ)∈F(M)×MF(E). We now fix (u, ψ)∈F(M)×MF(E) and let (ut, ψt)t∈R

be a horizontal curve in F(M)×M F(E) such that (u0, ψ0) = (u, ψ) and ( ˙u0,ψ˙0) = Hi(u, ψ). Letxt:=πut=πψt(so that ˙x0=uei). Then

//0,t=ψtψ⁻¹₀ :Ex→Ext

is the parallel transport (with respect to∇^E) along the curve (xt)t∈R. Consequently, (HiHjϑ)(u, ψ) =˜ d

dt t=0

ψ⁻¹_t ∇^E_utejϑ

=ψ₀⁻¹d dt

_t=0

//⁻¹_0,t∇^E_utejϑ

=ψ⁻¹Hessϑ(uei, uej),

as claimed.

Corollary 6.3 (Horizontal Laplacian, cf. [11, Proposition 3.1.2]).

For all(u, ψ)∈O(M)×F(E), we have

d

X

i=1

(H_i²ϑ)(u, ψ) =˜ ψ⁻¹∆ϑ(πψ).

Acknowledgements. This work has been supported by the Fonds National de la Recherche Luxembourg (OPEN Project GEOMREV).

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RobertPhilipowski

Mathematics Research Unit, University of Luxembourg, 6, rue Richard Coudenhove-Kalergi, L–1359 Luxembourg, Grand Duchy of Luxembourg

E-mail: [email protected]

AntonThalmaier

Mathematics Research Unit, University of Luxembourg, 6, rue Richard Coudenhove-Kalergi, L–1359 Luxembourg, Grand Duchy of Luxembourg

E-mail: [email protected]